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“Graded Commutative Algebra”. Our Techniques Will Be Different View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector MATHEMATICS Proceedings A 84 (3), September 28, 1981 About graded fields by J. van Gee1 and F. van Oystaeyen University of Antwerp, 2630 Wilrijk, Belgirrm Communicated by Prof. J.P. Murre at the meeting of September 27, 1980 INTRODUCTION In this paper we developwhat may be called “graded commutative algebra”. Our techniqueswill be different from the existing ones becausewe stay within the graded context i.e. we set up a theory about graded rings starting from the study of graded fields. The aim of this is the study of the (graded) Brauer group of a graded field and its relation to the Brauer groups of the quotient field and of the field of degree 0 in the graded field. Further results on these Brauer groups will be in a forthcoming paper, here in Section4 we give an introduction to this theory, including a graded version of Posner’s theorem for graded P.I. rings. The theory of algebraic graded extensions of graded fields given in Section 1 is useful in finding graded splitting fields of graded simple graded Artinian algebras.We have included graded versionsof Zariski’s theorem and Hilbert’s nulstellensatz;let us point out however that the graded nulstellensatz included here is different from the projective version of the theorem commonly used in projective geometry. Indeed, the affine “graded” geometry over graded fields is not a projective geometry over a common field (it can be shown that the affine geometry over a graded field may be considered as an hyperplane in Proj R for a certain graded ring R). Although-we do not go into the geometric theory here we do include our results on graded valuation rings in graded fields, but here mainly becauseof their usefulnessin determining integral closuresand graded integral closures. 273 Another advantage of the concept of working consequently in the graded context is that one may obtain short proofs for some known theorems with otherwise tedious proofs. For example, the well-known (and very useful in projective geometry) theorem stating that: the integral closure of a graded domain in its field of fractions is again a graded domain (cf. Theorem 11, [ 141) turns out to be a specialcase of a corollary to our results on integral closure and graded integral closure. The theory of graded valuation rings, cf. Section 3, yields an application of the theory of primes in rings expoundedby the authors in [7], [S] and in [lo], since a graded valuation ring in a graded field is a prime of that graded field. We assumeelementary facts about graded rings and refer to [S] for details and further references. In this paper all graded rings are Z-graded! 1. GRADED FIELDS All rings will be associative with units. Unless otherwise stated we will considercommutative rings. A graded ring R is said to be a graded division ring if every homogeneous element of R is invertible. A commutative graded division ring is called a graded field. The structure of a graded division ring is given in Theorem 4.1.) in the commutative casehowever it is easy to establish that a graded field K is either of the form Ko[t, t- ‘3 where KOis a field, the part of degree0 in K, and a variable given positive degree, or elseK = Ko. Let R be a graded field, let K be its field of fractions and let 52be an algebraic closure of K. An element a~ $2 is said to be graded algebraic over R if the minimal manic polynomial of a over K has the form: with aiER and deg (ai)-deg (ai+l)=cEH, for all i=O,...,n-1. The latter condition states exactly that f(X) is a homogeneouselement of R[X] equipped with the gradation RIXln = C i.+jcFn RiXj, n E H. A ring S containing R is said to be graded algebraic over R if S is graded and every s E S is graded algebraicover R. In thesedefinitions it may at first sight be restricting to demand that exactly the minimal manic polynomial over K is of the prescribedtype instead of merely demandingthe existenceof such a manic polynomial. Apart from the fact that this will not work, the following lemma justifies the definition we have chosen;with notations as above: LEMMA 1.1. If g(X) is a manic polynomial in R[X] with minimal degreein X such that g(a) = 0, then g(X) is a minimal polynomial for a over K. PROOF. If g(X) factorizes as hl(X)hz(X) in R[X], then the leading co- efficients of hl and hz are units in R, hencethey haveto be homogeneous.Since R is a graded field we may assumethat hi(X) and hz(X) are manic polynomials, but then the minimality assumption on degx g(X) yields that either hi or h2 equals 1, i.e. g(X) is irreducible in R[X]. Now R = Ro[ T, T-l] is a factorial ring and therefore g(X) will still be irreducible in K[X], cf. [4] p. 127. Consequently g(X) is a minimal polynomial for a over K. I 274 COROLLARY 1.2. An a E 52is graded algebraic over R if it is integral over R and the manic polynomial of minimal degree in X satisfied by a has the form (*). COROLLARY 1.3. If arz Q is graded algebraic over R then the ring R[a] (constructed within 9) is a graded field. PROOF. Let f(X), the minimal manic polynomial over R satisfied by a, be of the form (*). By the above lemma we have that R[a] zR[X]/(f(X)) as a ring. Moreover, if we equip R[X] with the gradation: R[X], = C i+jc=n RiXj, n E Z, then (j(X)) is a graded ideal of R[X] and this makes R[a] into a graded ring with deg (a) = C. If we establishthat cf(X)) is maximal as a graded ideal of R[X] then R[a] is a graded field. Therefore, consider a homogeneoush(X) E R[X]. The lemma entails that K[X](h(X), f(X)) equals either K[X] or K[X]df(X)). In the first casethe graded ideal u(X)), h(X)) containsa homogeneouselement of R i.e. an unit of R, thus in this case(h(X), f(X)) = R[X]. In the secondcase there exists an r ER such that r/r(X) E cf(X)), consequently:r&(X) E (j(X)) for each homo- geneouscomponent ri of r. SinceR is a graded field h(X) E (j(X)) follows. Thus (j(X)) is a maximal graded ideal of R[X]. H LEMMA I .4. Let a E Q be graded algebraicover R and of degreec,, let PE 52 be graded algebraic over R and of degreecp, then p is graded algebraic over R[a] and of degreeCD. PROOF. Let h be a manic polynomial for j? over R[a] and supposethat h has minimal degree in X as such. Since R[a] is a graded field (Corollary 1.3.), R[a][X] is a factorial ring and thus h is irreducible (Lemma I. 1.) and also h is the minimal polynomial for fl over K(a) (Lemma 1.1. again), where K(a) denotes the field of fractions of R[a]. Let gf R[X] be the minimal manic polynomial for j? over R, then g is homogeneousin R[X] if the latter ring is graded by putting R[X],, = Ci+jcszn RiXi. We embedR[X] into R[a][X] in the obvious way. In R[a][X] we have then that g = h< + p with p(J) = 0, hencep = 0 follows by the minimality condition on degx h. The ring R[a] [X] is graded such that deg (X) = cg, thus g = ht implies that h and { are homogeneousin R[a][X] i.e. h has the desiredform (A) over R[a]. Consequentlyp is graded algebraic over R[a] of degreeC,T. n COROLLARY 1.5. The set of graded algebraicelements over R within 52is a graded field. PROOF. Put E(R)c = (a E 0, a graded algebraic over R, deg (a) = c}. Take a,/? EE(R)c and considerR[a, p] in a. From the foregoing results it follows that R[a,j?] is a graded field, thus cr+p is homogeneousof degreec in R[a,/?]. If C r=-, ri(o + /?)’ is the minimal manic polynomial for a + p over R then by 275 selectingterms of highest degree,N say, we obtain: j+,lN %(a +P)‘=O where ri,j is the homogeneouscomponent of degreej of r;. Up to multiplication by a unit of R we may assumethat f(X) = Cj+ic=N QjX’is a manic polynomial of the form (*) which is satisfied by o+ #I, hence a+ DE JY(R)~. In much the same way it may be established that, (xEEL and beE( yields C@EE(R),+d. Finally, it follows that E(R) = @cE~E(R)c is a graded ring, evena graded field. n REMARK 1.6. E(R) is termed to be the graded algebraic closure of R in 52. The mere existenceof a graded algebraic closure could have been established more directly, as in the ungraded case. LEMMA 1.7. Every graded module over a graded field is graded free i.e. has a basisconsisting of homogeneouselements. PROOF. Easy. H As a consequenceof the above lemma we may talk about the (graded) rank of a graded module M over a graded field, we will denote it by [A&R].
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